Volume 46, Issue 11, 01 June 1967

Numerical Determination of Chemical Equilibrium and the Partitioning of Free Energy
View Description Hide DescriptionA simplified procedure is developed for the numerical determination of the equilibrium composition of complex mixtures. Computational steps have been simplified and numerical difficulties arising in previous methods are eliminated. A new concept leading to the allocation of free energy to the atoms or elementary particles comprising a molecule or other composite structure is suggested. Numerical values are derived for the free energy of hydrogen and oxygen atoms in a water molecule.

Microwave Spectrum of Allyl Alcohol
View Description Hide DescriptionThe microwave spectrum of allyl alcohol has been investigated in the frequency range 8–35 Gc/sec, for the normal sample (CH_{2}CHCH_{2}OH), deuterated sample (CH_{2}CHCH_{2}OD), and ^{18}O‐substituted sample (CH_{2}CHCH_{2} ^{18}OH). For the normal sample, 20 transitions, which can be attributed to rotational absorption by a gauche form of the molecule, are reported both for the ground vibrational state and the first excited vibrational state. The rotational constants of the deuterated sample and ^{18}O‐substituted sample indicate the assigned conformation is the gauche form which allows an H bond between the π electrons and the OH group. The molecular‐dipole‐moment components have been determined from the Stark‐effect measurements, μ_{ a }=1.41±0.02 D, μ_{ b }=0+0.07 D, μ_{ c }=0.65±0.01 D, and μ_{tot}=1.55±0.04 D.

Seventh Virial Coefficients for Hard Spheres and Hard Disks
View Description Hide DescriptionThe seventh virial coefficient B _{7} is expressed as a sum of modified star integrals instead of the usual Mayer star integrals. The new graphs contain both Mayer f functions and f̃(≡1+f) functions. This simplifies the calculation of B _{7}. That is, instead of the 468 star integrals that appear in Mayer's formulation, only 171 modified star integrals now appear in the evaluation of B _{7}. Furthermore, for D‐dimensional particles with a hard core, these 171 integrals are strongly dependent on the number of dimensions D. For hard rods, hard disks, and hard spheres, respectively, at most one, 78, and 164 of these integrals give a nonvanishing contribution to B _{7}. When Monte Carlo integration is used to evaluate these integrals using hard‐sphere and hard‐disk potentials we obtain the following values of B _{7}: hard spheres, B _{7}/(B _{2})^{6}=0.0138±0.0004; hard disks, B _{7}/(B _{2})^{6}=0.1141±0.0005. For hard spheres, the truncated seven‐term virial series for the pressure agrees within 10% with the results of the molecular dynamics data throughout the fluid phase. The Padé approximants to the pressure obtained from B_{n} (n<8) data agree even better (≲5%) with the molecular‐dynamics data on the fluid side. Estimated values of B _{8}/(B _{2})^{7} obtained from the Padé approximants are 0.005 and 0.063 for hard spheres and hard disks, respectively. Furthermore, for hard disks, the Padé approximants obtained from the seven known values of B_{n} (n<8) indicate that the virial series shows an unstable phase at ρ=ρ_{0}. If this is in fact true, then the low‐density study of the equation of state for hard disks would yield some information on the high‐density side. We have also calculated the first seven coefficients C_{n} of the density expansion of the isothermal compressibility for hard spheres, hard disks, hard parallel cubes, and hard parallel squares. We find that these C_{n} alternate in sign. For two‐dimensional continuum particles with a hard core, the corresponding compressibility series appear to converge faster than the usual virial series of the pressure. We also compare the present values of B _{7} with the various values of B _{7} calculated from approximate theories.

Solution Properties of Synthetic Polypeptides. III. Viscosity Behavior of Poly‐γ‐Benzyl‐L‐Glutamate in the Helix—Coil Transition Region
View Description Hide DescriptionLimiting viscosity numbers [η] and specific rotations [α]_{578} of five samples of poly‐γ‐benzyl‐L‐glutamate (PBLG) (ranging in weight‐average molecular weight M̄_{w} from 1.9×10^{4} to 4.0×10^{5}) in mixtures of dichloroacetic acid (DCA) and ethylene dichloride (EDC) of various compositions were measured as functions of temperature in the range where the PBLG molecule underwent a thermal transition from a disordered, random‐coil state to an ordered helical state. For one sample the measurements of optical rotatory dispersion were made at various solvent compositions and temperatures, and the data were analyzed in terms of the Moffitt plot. When the M̄_{w} of a given sample was above a certain critical value, [η] first decreased (or was kept constant), passed through a shallow minimum, and then increased sharply to a limiting value as the temperature was raised. Below the critical value, the change in [η] with temperature was a monotonic decrease. Irrespective of the molecular weight of the sample, [α]_{578} increased sharply in a narrow range near 30°C and then gradually leveled off as the temperature was raised. This narrow region coincided with the region where [η] underwent a marked increase. It was found that the helical content f _{H} of a given sample can be evaluated from the reduced mean residue rotation measured at 578 mμ. When the values of [η] obtained under different solvent compositions and temperatures are plotted against the f _{H} values so calculated, the plotted points formed a single composite curve over the entire range of f _{H}. This curve exhibits a quite similar character to the theoretical curves for 〈S ^{2}〉 (mean‐square radius of gyration) vs f _{H} calculated by Nagai on the statistical‐mechanical treatment of helix‐forming polypeptides.

Shock Compression of Argon. II. Nonadditive Repulsive Potential
View Description Hide DescriptionBy means of the Monte Carlo method the shock Hugoniots of argon have been analyzed for an effective pair potential. This effective pair potential for the low‐density Hugoniots closely agrees with the potential derived from molecular beams indicating the high accuracy of the pair‐wise additivity approximation besides confirming the consistency of two complex experiments. For the high‐density Hugoniot, however, the effective pair potential is considerably stiffer than the pair‐wise potential at high compression. For liquid argon compressed twofold, the nonadditivity effect in the potential is about 30%. The theoretically derived pair‐wise potential is only a few percent softer than this nonadditive potential. This theory might be improved further by allowing for distortions of the outer electron shell in a direction away from the contact region of the interacting pair of atoms. At high density the surrounding medium inhibits this distortion which leads to the observed stiffening and to the close agreement with the theory of spherical atoms.

Generalized Diffusion Theory of Nonequilibrium Reaction Rates
View Description Hide DescriptionA simple expansion procedure is used in conjunction with the detailed balancing condition to show that the linear master equation describing the relaxation of an atomic system in a background of inert atoms can be approximated by an equivalent diffusionequation with appropriate boundary conditions. This equation has the form of the ordinary heat‐conduction equation for a medium of variable heat capacity and thermal conductivity and is simplier than the commonly used Fokker—Planck equation. It is expected to be a reasonable approximation under conditions where the changes in the constants of the motion are small in a single collision. A similar equation has been derived by Van Kampen who showed that it implies the phenomenological equations of nonequilibrium thermodynamics. However, he did not consider the boundary condiditions necessary to deal with chemical reactions or phase changes.

Acetylene—Oxygen Reaction in Shock Waves. Origin of CO_{2}
View Description Hide DescriptionThe ir emissions from CO and CO_{2}, formed in the C_{2}H_{2}–O_{2}reaction in a shock tube, have been observed as a function of time. By calibrating these emissions in terms of concentrations and by comparing them with the emission of CH^{*} (A ^{2}Δ→X ^{2}π), produced in the same reaction, it has been possible to confirm that CO_{2} is formed in the early stages by a reaction other than CO+OH→CO_{2}+H. This reaction involves a primary carbon‐containing radical and an O_{2} molecule, and its rate at 1800°K is 5% of the rate of formation of CO, produced by a similar reaction.
The importance of the reaction CO+OH→CO_{2}+H in the later stages of oxidation has been examined as a function of the initial O_{2}/C_{2}H_{2} ratio, and the growth of OH concentration during the reaction elucidated.
More decisive proof is given that CH^{*} is produced by a reaction which is second order in reaction intermediates.

Atomic r_{ij} Integrals for Many‐Electron Wavefunctions
View Description Hide DescriptionA general integration scheme is developed for doing multielectron r_{ij} integrations. Preliminary calculations on Be indicate that it is practical to obtain variational wavefunctions containing unliked four‐electron correlated terms up through at least neon by using this method.

Vibrational Energy Transfer in CO_{2} Lasers
View Description Hide DescriptionLaser‐excited vibrational fluorescencemeasurements have been made on the asymmetric‐stretching vibrational level (00°1) of CO_{2}. Vibration→vibration energy‐transfer rates from this level due to collisions with CO_{2} and with a number of other collision partners are presented. The rate of near‐resonant exchange of vibrational energy between CO_{2} and N_{2} (ΔE=18 cm^{−1}) has been measured. The kinetics of the CO_{2} laser system are analyzed in terms of a three‐level scheme. Observed laser performance is compared with that calculated by use of collisional and radiative coupling rates observed in nonionized gases and of electron activation and deactivation rates estimated from CO_{2} discharge systems. In accordance with the scheme presented, the relative effectiveness of small amounts of added H_{2}, D_{2}, and He on laser output parallels their effectiveness in deactivating the lower laser level. The criteria for selecting molecules with vibrational‐energy‐level patterns likely to produce laser systems are outlined. Attempts to produce new continuous‐wave lasers in SO_{2} and HCN are described.

Absorption Spectra of Pr^{3+} and Nd^{3+} Doped in Lanthanum Sulfate
View Description Hide DescriptionThe crystal‐field parameters of lanthanum sulfate are calculated by studying the absorptionspectrum of Pr^{3+}‐ and Nd^{3+}‐doped lanthanum sulfate. These values are compared with the previously obtained parameters for lanthanum ethyl sulfate. It is shown that A _{2} ^{0}〈r ^{2}〉 is quite large in the sulfate in comparison with the ethyl sulfate.

Electronic Excited States of Benzene and Ethylene
View Description Hide DescriptionEnergies of low‐lying excited electronic states of benzene have been calculated using the semiempirical SCF scheme of Pople, Santry, and Segal. Semiquantitatively transferable parameters are chosen by fitting the theory to the ionization potential and the position of the ^{3} B _{1u } state. The principal requisite modification of parameters concerns a reduction in scale of resonance integrals from those suggested by Pople et al.; subsidiary small adjustments are required in diagonal core parameters.
The theory predicts several interesting features which are at variance with the usual interpretation of the near‐ultraviolet spectrum but which are not in irrevocable conflict with experiment. When these are taken into account, positions of other states, calculated without scaling of atomic‐orbital Coulomb repulsion integrals, are in good agreement with experiment.
Representative results are also presented for ethylene.

Chlorine and Deuteron Relaxation Time Measurements in Liquid HCl and DCl
View Description Hide DescriptionMeasurements of chlorine T _{2} in liquid HCl and DCl have been made over the temperature range of −40° to 60°C. Estimates of the reorientational correlation time made possible by the chlorine measurements were used to evaluate the proton intramolecular dipole contribution which was subtracted from the total intramolecular contribution to obtain the proton spin—rotational contribution. The magnitude of the spin—rotational contribution thus obtained compares with that estimated by Krynicki and Powles.
Deuteron T _{1} times in liquid DCl were measured over the temperature range of −75° to 25°C and found to differ considerably both in magnitude and temperature dependence from deuteron T _{2} times. Estimates of the quadrupole contribution to the deuteron (1/T _{1}) and, thus, a value of 2.0×10^{17} V/cm^{2} for the electric‐field gradient at the deuteron site in DCl were made by use of the reorientational correlation time from chlorine measurements.

Temperature Dependence of Singlet—Triplet Intersystem Crossing in Anthracene Crystals
View Description Hide DescriptionThe formation of triplet excitons from singlet excitons via a radiationless transition (intersystem crossing) is studied in anthracene crystals. Using blue exciting light with 430 mμ>λ>350 mμ the temperature dependence of the intersystem crossing is found to follow an exponential law with an activation energy of 800±50 cm^{−1}.
It is concluded that this transition starts from a vibrationally excited single state and involves the second triplet state. The total energy of this vibrational state is about 26 100 cm^{−1} in good agreement with that suggested by Kellogg, 26 050 cm^{−1}, for an excited triplet level.
No emission from the triplet—triplet transition could be observed.

Diffusion Length of Triplet Excitons in Anthracene Crystals
View Description Hide DescriptionRecently two different experimental principles have been used to determine the diffusion length of triplet excitons and the results have differed by an order of magnitude. Experiments have now been performed using (i) singlet excitation and (ii) triplet lifetime dependence on thickness, and the results are shown to be in accord with results from other techniques. The triplet diffusion length l is shown to be 20±10 μ. Also it is found that l is slightly temperature dependent, with an apparent activation energy of approximately 10^{−2} eV.

Photoelectron Spectra of the Halogens and the Hydrogen Halides
View Description Hide DescriptionA spherical‐grid photoelectron spectrometer has been used to measure the ionization potentials of HF, HCl, HBr, HI, F_{2}, Cl_{2}, Br_{2}, and I_{2}. The relative transition probabilities to several of the ionic states are given. There is evidence that the ^{2}Σ^{+} ionic states in HI^{+}, Cl_{2} ^{+}, Br_{2} ^{+}, and I_{2} ^{+} are unstable.

NMR Studies of a Series of Aliphatic Nitroxide Radicals
View Description Hide DescriptionWe have examined the NMR and ESR spectra of a series of aliphatic nitroxide radicals. Isotropic coupling constants were calculated from the shifts of the NMR lines. The NMR shifts were investigated as a function of solvent and temperature. Coupling constants determined from NMR spectra were used to calculate ESR spectra. The experimental and theoretical ESR spectra were in good agreement.

Theoretical Investigations of Gas—Solid Interaction Phenomena. I
View Description Hide DescriptionA classical study of gas—solid surface interactions is reported. A two‐dimensional, independent oscillator lattice (IOL) is used as a surface model with harmonic potentials connecting each lattice site. Morse‐type potential functions are assumed to operate between the incident gaseous atom and each lattice atom. Using Monte Carlo techniques and numerical integration of the 16 differential equations describing the system, energy accommodation coefficients (EAC) and spatial distributions have been calculated for a (He/Ni) system. The dependence of these quantities upon incidence angle, gaseous beam temperature, surface temperature, and various interaction parameters has been investigated. The results indicate that the EAC increases with increasing beam temperature and beam velocity; decreases with increasing lattice force constant; increases with increasing attractive interaction between gas and surface; decreases with increasing incidence angle; and decreases as the surface and gaseous temperatures approach one another. In general, these results are in qualitative to semiquantitative agreement with experiment. The calculated spatial distributions depend strongly upon incidence angle. For perpendicular incidence (θ_{ i }=0°) a diffuse distribution is predicted. For θ_{ i }=37.5° the distribution is peaked at an angle subspecular by 18°. As the attractive interactions between gas and surface increase, the maximum is shifted toward the surface normal. Similar behavior is observed with respect to an increase in surface temperature. Comparison with experiment indicates the He scattering to be specular rather than 18° subspecular. The calculated distributions are also too diffuse due probably to the two‐dimensional character of the model. The predicted trends with regard to attractive interactions and surface temperature are in accord with observation. The over‐all behavior indicates that the ``normal‐component model'' of energy transfer is oversimplified and that transfer of energy in the parallel component of motion can be of importance.

Mass‐Spectrometric and Theoretical Evidence for NH_{4} and H_{3}O
View Description Hide DescriptionThe unusual chemical species H_{3}O and NH_{4} have been studied using both experimental and theoretical methods. Experimentally the species were investigated by means of a mass spectrometer equipped with two different reactors designed to produce reactive species. The NH_{4} was produced by surface chemistry techniques, whereas H_{3}O was produced by irradiatingwater vapor with ionizing electrons. A study of the ionization potential of NH_{4} by surface ionization techniques gave a value of 5.9 eV. The ionization potential of H_{3}O was measured by conventional techniques and a value of 10.9 eV was obtained. In the theoretical investigation, the physical parameters are given including s‐, p‐, and d‐type orbitals for H_{3}O and s‐, p‐, and f‐type orbitals for NH_{4}. The calculations for NH_{4} predict a tetrahedral structure with internuclear distances of about 1.06 Å and they predict a planar H_{3}O molecule with bond distances about 1.03 Å. The ionization potentials of H_{3}O and NH_{4} were estimated to 3.9 and 3.0 eV.

Frenkel and Wannier Spin Excitons in Organic Free‐Radical Crystals
View Description Hide DescriptionIt is shown that organic molecular crystals containing linear chains of exchange‐coupled free radicals fall into two classes which may be characterized by their spin excitations. Two spins forming a Frenkel spin exciton remain adjacent in the crystal, while spins forming a Wannier spin exciton move through the crystal independently. This difference in the structure of the triplet excitations leads to a different temperature and pressure dependence for the width of the exchange‐narrowed line observed by spin resonance, to different exciton fine structure, and to different magnetic susceptibility‐vs‐temperature curves. It is suggested that paramagnetic excitations in organic charge‐transfercompounds containing chains of alternately positive and negative ion radicals are Wannier spin excitons.

Overlap as a Criterion for Conduction in the Cubic Tungsten Bronzes
View Description Hide DescriptionOverlap calculations using Slater orbitals with exponents adjusted to match SCF functions have been performed on eight pairs of orbitals which have been considered to have some pertinence to conduction in the cubic sodiumtungsten bronzes. The results support the dπ—pπ model proposed by Sienko and by Good‐enough and clearly indicate where the overlap criterion alone breaks down, namely for the case of sodium—sodium pσ—pσ overlap.